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Topics

1. basic topological notions
2. examples of topological spaces
3. separation properties
4. classification of 2−manifolds
5. introduction to homotopy theory

Outline/Homework

• Mon August 25: introduction, definition of topology
• Wed August 27: basis for a topology, interior, closure, limit points
• Fri August 29: metric spaces, metric topologies
• HW 1: due Wednesday, September 3 (L^aT_eX template) SOLUTION
• Mon September 1: No Class (Labor Day)
• Wed September 3: continuous mappings
• Fri September 5: subspaces
• HW 2: due Wednesday, September 10 (L^aT_eX template) SOLUTION
• Mon September 8: product spaces
• Wed September 10: connectedness
• Fri September 12: connected components and local connectedness
• HW 3: due Wednesday, September 17 (L^aT_eX template) SOLUTION
• Mon September 15: path connectedness and path components
• Wed September 17: compactness
• Fri September 19: compactness continued, finite intersection property
• HW 4: due Wednesday, September 24 (L^aT_eX template) SOLUTION
• Mon September 22: Tychonoff theorem
• Wed September 24: function spaces
• Fri September 26: separation axioms
• HW 5: due Wednesday, October 1 (L^aT_eX template) SOLUTION
• Mon September 29: Hausdorff, regular and normal spaces
• Wed October 1: Urysohn's lemma
• Fri October 3: Tietze extension theorem
• HW 6: due Wednesday, October 8 (L^aT_eX template) SOLUTION
• Mon October 6: Urysohn's metrization theorem
• Wed October 8: Complete metric spaces, Baire spaces
• Fri October 10: Peano's space filling curve
• HW 7: due Wednesday, October 15 (L^aT_eX template) SOLUTION
• Mon October 13: inverse systems
• Wed October 15: characterization of the Cantor set
• Fri October 17: characterization of the Cantor set continued
• HW 8: due Wednesday, October 29 (L^aT_eX template) SOLUTION
• Mon October 20: No Class (Fall Break)
• Wed October 22: quotient spaces
• Fri October 24: manifolds
• Mon October 27: embeddings of manifolds
• Wed October 29: simplicial and PL complexes
• Fri October 31: mapping class groups (Matt Durham)
• HW 9: due Wednesday, November 5 (L^aT_eX template) SOLUTION
• Mon November 3: polygons in R²
• Wed November 5: Schönflies theorem for polygons in R²
• Fri November 7: Jordan curve theorem
• HW 10: due Monday, November 17 (L^aT_eX template) SOLUTION
• Mon November 10: Jordan curve theorem continued
• Wed November 12: Euler characteristic
• Fri November 14: ??
• Mon November 17: classification of compact, connected 2−manifolds
• Wed November 19: classification of compact, connected 2−manifolds continued
• Fri November 21: classification of compact, connected 2−manifolds continued
• HW 11: due Monday, December 1 (L^aT_eX template) SOLUTION
• Mon November 24: introduction to homotopy
• Wed November 26: No Class (Thanksgiving Holiday)
• Fri November 28: No Class (Thanksgiving Holiday)
• Mon December 1: criteria for homotopy equivalence
• Wed December 3: homotopy extension property
• Fri December 5: the fundamental group
• HW 12: due Wednesday, December 10 (L^aT_eX template)
• Mon December 8: applications of the fundamental group
• Wed December 10: the induced homomorphism
• Mon December 15: FINAL EXAM

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